/*  
  A C version of Kahan's Floating Point Test "Paranoia"

      Thos Sumner, UCSF, Feb. 1985
      David Gay, BTL, Jan. 1986

  This is a rewrite from the Pascal version by

      B. A. Wichmann, 18 Jan. 1985

  (and does NOT exhibit good C programming style).

  Adjusted to use Standard C headers 19 Jan. 1992 (dmg);

(C) Apr 19 1983 in BASIC version by:

  Professor W. M. Kahan,
  567 Evans Hall
  Electrical Engineering & Computer Science Dept.
  University of California
  Berkeley, California 94720
  USA

converted to Pascal by:

  B. A. Wichmann
  National Physical Laboratory
  Teddington Middx
  TW11 OLW
  UK

converted to C by:

  David M. Gay 
  AT&T Bell Labs
  600 Mountain Avenue
  Murray Hill, NJ 07974, USA

and  

  Thos Sumner
  Computer Center, Rm. U-76
  University of California
  San Francisco, CA 94143, USA

with simultaneous corrections to the Pascal source (reflected
in the Pascal source available over netlib).
[A couple of bug fixes from dgh = sun!dhough incorporated 31 July 1986.]

Reports of results on various systems from all the versions
of Paranoia are being collected by Richard Karpinski at the
same address as Thos Sumner.  This includes sample outputs,
bug reports, and criticisms.

You may copy this program freely if you acknowledge its source.
Comments on the Pascal version to NPL, please.


The C version catches signals from floating-point exceptions.
If signal(SIGFPE,...) is unavailable in your environment, you may
#define NOSIGNAL to comment out the invocations of signal.

By #defining Single when you compile this source, you may obtain
a single-precision C version of Paranoia.
*/

#include <stdio.h>

#ifndef NOSIGNAL
#include <signal.h>
#endif

#include <setjmp.h>

#ifdef Single
# define FLOAT float
# define FABS(x) (float)fabs((double)(x))
# define FLOOR(x) (float)floor((double)(x))
# define LOG(x) (float)log((double)(x))
# define POW(x,y) (float)pow((double)(x),(double)(y))
# define SQRT(x) (float)sqrt((double)(x))
#else
# define FLOAT double
# define FABS(x) fabs(x)
# define FLOOR(x) floor(x)
# define LOG(x) log(x)
# define POW(x,y) pow(x,y)
# define SQRT(x) sqrt(x)
#endif

jmp_buf ovfl_buf;
#ifdef __STDC__
#include <stdlib.h>
#include <math.h>
#else

#ifdef __cplusplus
extern "C" {
#endif

extern double fabs(double);
extern double floor(double);
extern double log(double);
extern double pow(double,double);
extern double sqrt(double);
extern void exit( int );

#ifdef __cplusplus
  }
#endif

#endif
typedef void (*Sig_type)(int);

void sigfpe ( int x );
FLOAT Sign(FLOAT), Random(void);
extern void BadCond(int, char*);
extern void SqXMinX(int);
extern void TstCond(int, int, char*);
extern void notify(char*);
extern int read(int, char*, int);

#undef V9
extern void Characteristics( void );
extern void Heading( void );
extern void History( void );
extern void Instructions( void );
extern void IsYeqX( void );
extern void NewD( void );
extern void Pause( void );
extern void PrintIfNPositive( void );
extern void SR3750( void );
extern void SR3980( void );
extern void TstPtUf( void );

Sig_type sigsave;

#define KEYBOARD 0

FLOAT Radix;
FLOAT RadixD2;
FLOAT BInvrse, BMinusU2;

/*
  Small floating point constants.
*/
FLOAT Zero = 0.0;
FLOAT Half = 0.5;
FLOAT One = 1.0;
FLOAT Two = 2.0;
FLOAT Three = 3.0;
FLOAT Four = 4.0;
FLOAT Five = 5.0;
FLOAT Eight = 8.0;
FLOAT Nine = 9.0;
FLOAT TwentySeven = 27.0;
FLOAT ThirtyTwo = 32.0;
FLOAT TwoForty = 240.0;
FLOAT MinusOne = -1.0;
FLOAT OneAndHalf = 1.5;
/*
  Integer constants
*/
int NoTrials = 20; /*Number of tests for commutativity. */
#define False 0
#define True 1

/* 
  Definitions for declared types 
  Guard == (Yes, No);
  Rounding == (Chopped, Rounded, Other);
  Message == packed array [1..40] of char;
  Class == (Flaw, Defect, Serious, Failure);
*/
#define Yes 1
#define No  0
#define Chopped 2
#define Rounded 1
#define Other   0
#define Flaw    3
#define Defect  2
#define Serious 1
#define Failure 0
typedef int Guard, Rounding, Class;
typedef char Message;

/* 
  Declarations of Variables 
*/
int Indx;
char ch[8];
FLOAT AInvrse, A1;
FLOAT C, CInvrse;
FLOAT D, FourD;
FLOAT E0, E1, Exp2, E3, MinSqEr;
FLOAT SqEr, MaxSqEr, E9;
FLOAT Third;
FLOAT F6, F9;
FLOAT H, HInvrse;
int I;
FLOAT StickyBit, J;
FLOAT MyZero;
FLOAT Precision;
FLOAT Q, Q9;
FLOAT R, Random9;
FLOAT T, Underflow, S;
FLOAT OneUlp, UfThold, U1, U2;
FLOAT V, V0, V9;
FLOAT W;
FLOAT X, X1, X2, X8, Random1;
FLOAT Y, Y1, Y2, Random2;
FLOAT Z, PseudoZero, Z1, Z2, Z9;
int ErrCnt[4];
int fpecount;
int Milestone;
int PageNo;
int M, N, N1;
Guard GMult, GDiv, GAddSub;
Rounding RMult, RDiv, RAddSub, RSqrt;
int Break, Done, NotMonot, Monot, Anomaly, IEEE,
    SqRWrng;
int UfNGrad;
/* 
  Computed constants. 
*/
/*U1  gap below 1.0, i.e, 1.0-U1 is next number below 1.0 */
/*U2  gap above 1.0, i.e, 1.0+U2 is next number above 1.0 */

/**********************************************************************/

int main ( void )

/**********************************************************************/
/*
  Purpose:

    PARANOIA tests the floating point arithmetic implementation on a computer.
*/
{
/*
  First two assignments use integer right-hand sides.
*/
  Zero = 0;
  One = 1;
  Two = One + One;
  Three = Two + One;
  Four = Three + One;
  Five = Four + One;
  Eight = Four + Four;
  Nine = Three * Three;
  TwentySeven = Nine * Three;
  ThirtyTwo = Four * Eight;
  TwoForty = Four * Five * Three * Four;
  MinusOne = -One;
  Half = One / Two;
  OneAndHalf = One + Half;
  ErrCnt[Failure] = 0;
  ErrCnt[Serious] = 0;
  ErrCnt[Defect] = 0;
  ErrCnt[Flaw] = 0;
  PageNo = 1;
/*=============================================*/
  Milestone = 0;
/*=============================================*/

#ifndef NOSIGNAL
  signal ( SIGFPE, sigfpe );
#endif

  Instructions();
  Pause();
  Heading();
  Pause();
  Characteristics();
  Pause();
  History();
  Pause();
/*=============================================*/
  Milestone = 7;
/*=============================================*/

  printf ( "Program is now RUNNING tests on small integers:\n" );
  
  TstCond ( Failure, (Zero + Zero == Zero), "0+0 != 0" );
  TstCond ( Failure, (One - One == Zero),   "1-1 != 0" );
  TstCond ( Failure, (One > Zero),          "1 <= 0"   );
  TstCond ( Failure, (One + One == Two),    "1+1 != 2" );

  Z = - Zero;

  if ( Z != 0.0 )
  {
    ErrCnt[Failure] = ErrCnt[Failure] + 1;
    printf ( "Comparison alleges that -0.0 is Non-zero!\n" );
    U2 = 0.001;
    Radix = 1;
    TstPtUf();
  }

  TstCond ( Failure, (Three == Two + One),           "3 != 2+1" );
  TstCond ( Failure, (Four == Three + One),          "4 != 3+1" );
  TstCond ( Failure, (Four + Two * (- Two) == Zero), "4+2*(-2) != 0" );
  TstCond ( Failure, (Four - Three - One == Zero),   "4-3-1 != 0" );

  TstCond (Failure, (MinusOne == (0 - One))
       && (MinusOne + One == Zero ) && (One + MinusOne == Zero)
       && (MinusOne + FABS(One) == Zero)
       && (MinusOne + MinusOne * MinusOne == Zero),
       "-1+1 != 0, (-1)+abs(1) != 0, or -1+(-1)*(-1) != 0");

  TstCond (Failure, Half + MinusOne + Half == Zero,
      "1/2 + (-1) + 1/2 != 0");

/*=============================================*/
  Milestone = 10;
/*=============================================*/

  TstCond (Failure, (Nine == Three * Three)
       && (TwentySeven == Nine * Three) && (Eight == Four + Four)
       && (ThirtyTwo == Eight * Four)
       && (ThirtyTwo - TwentySeven - Four - One == Zero),
       "9 != 3*3, 27 != 9*3, 32 != 8*4, or 32-27-4-1 != 0");

  TstCond (Failure, (Five == Four + One) &&
      (TwoForty == Four * Five * Three * Four)
       && (TwoForty / Three - Four * Four * Five == Zero)
       && ( TwoForty / Four - Five * Three * Four == Zero)
       && ( TwoForty / Five - Four * Three * Four == Zero),
      "5 != 4+1, 240/3 != 80, 240/4 != 60, or 240/5 != 48");

  if (ErrCnt[Failure] == 0)
  {
    printf ( "-1, 0, 1/2, 1, 2, 3, 4, 5, 9, 27, 32 & 240 are O.K.\n" );
    printf ( "\n");
  }

  printf ( "Searching for Radix and Precision.\n" );
  W = One;

  do
  {
    W = W + W;
    Y = W + One;
    Z = Y - W;
    Y = Z - One;
  } while (MinusOne + FABS(Y) < Zero);

/*
  Now W is just big enough that |((W+1)-W)-1| >= 1.
*/
  Precision = Zero;
  Y = One;
  do
  {
    Radix = W + Y;
    Y = Y + Y;
    Radix = Radix - W;
  } while ( Radix == Zero);

  if ( Radix < Two ) 
  {
    Radix = One;
  }

  printf ( "Radix = %f\n", Radix );

  if (Radix != 1)
  {
    W = One;
    do
    {
      Precision = Precision + One;
      W = W * Radix;
      Y = W + One;
    } while ((Y - W) == One);
  }
/*
  Now W == Radix^Precision is barely too big to satisfy (W+1)-W == 1
*/
  U1 = One / W;
  U2 = Radix * U1;
  printf ( "Closest relative separation found is U1 = %.7e\n", U1 );
  printf ( "\n" );
  printf ( "Recalculating radix and precision\n" );
/*
  save old values
*/
  E0 = Radix;
  E1 = U1;
  E9 = U2;
  E3 = Precision;
  
  X = Four / Three;
  Third = X - One;
  F6 = Half - Third;
  X = F6 + F6;
  X = FABS(X - Third);
  if (X < U2) X = U2;
/*
  now X = (unknown no.) ulps of 1+
*/
  do  {
    U2 = X;
    Y = Half * U2 + ThirtyTwo * U2 * U2;
    Y = One + Y;
    X = Y - One;
    } while ( ! ((U2 <= X) || (X <= Zero)));  
/*
  now U2 == 1 ulp of 1 +
*/
  X = Two / Three;
  F6 = X - Half;
  Third = F6 + F6;
  X = Third - Half;
  X = FABS(X + F6);

  if ( X < U1 )
  {
    X = U1;
  }
/*
  now  X == (unknown no.) ulps of 1 -
*/
  do  
  {
    U1 = X;
    Y = Half * U1 + ThirtyTwo * U1 * U1;
    Y = Half - Y;
    X = Half + Y;
    Y = Half - X;
    X = Half + Y;
  } while ( ! ((U1 <= X) || (X <= Zero)));
/*
  now U1 == 1 ulp of 1 -
*/
  if ( U1 == E1 )
  {
    printf ( "confirms closest relative separation U1 .\n" );
  }
  else 
  {
    printf ( "gets better closest relative separation U1 = %.7e .\n", U1 );
  }

  W = One / U1;
  F9 = (Half - U1) + Half;
  Radix = FLOOR(0.01 + U2 / U1);

  if ( Radix == E0 ) 
  {
    printf ( "Radix confirmed.\n" );
  }
  else 
  {
    printf ( "MYSTERY: recalculated Radix = %.7e .\n", Radix );
  }

  TstCond (Defect, Radix <= Eight + Eight,
       "Radix is too big: roundoff problems");
  TstCond (Flaw, (Radix == Two) || (Radix == 10)
       || (Radix == One), "Radix is not as good as 2 or 10");
/*=============================================*/
  Milestone = 20;
/*=============================================*/
  TstCond (Failure, F9 - Half < Half,
       "(1-U1)-1/2 < 1/2 is FALSE, prog. fails?");
  X = F9;
  I = 1;
  Y = X - Half;
  Z = Y - Half;
  TstCond (Failure, (X != One)
       || (Z == Zero), "Comparison is fuzzy,X=1 but X-1/2-1/2 != 0");
  X = One + U2;
  I = 0;
/*=============================================*/
  Milestone = 25;
/*=============================================*/
  /*... BMinusU2 = nextafter(Radix, 0) */
  BMinusU2 = Radix - One;
  BMinusU2 = (BMinusU2 - U2) + One;
/* 
  Purify Integers.
*/
  if (Radix != One)
  {
    X = - TwoForty * LOG(U1) / LOG(Radix);
    Y = FLOOR(Half + X);
    if (FABS(X - Y) * Four < One) X = Y;
    Precision = X / TwoForty;
    Y = FLOOR(Half + Precision);
    if (FABS(Precision - Y) * TwoForty < Half) Precision = Y;
  }

  if ((Precision != FLOOR(Precision)) || (Radix == One))
  {
    printf("Precision cannot be characterized by an Integer number\n");
    printf("of significant digits but, by itself, this is a minor flaw.\n");
  }

  if (Radix == One) 
  {
    printf("logarithmic encoding has precision characterized solely by U1.\n");
  }
  else
  {
    printf("The number of significant digits of the Radix is %f .\n", Precision);
  }

  TstCond (Serious, U2 * Nine * Nine * TwoForty < One,
       "Precision worse than 5 decimal figures  ");

/*=============================================*/
  Milestone = 30;
/*=============================================*/
  /* Test for extra-precise subepressions */
  X = FABS(((Four / Three - One) - One / Four) * Three - One / Four);
  do  {
    Z2 = X;
    X = (One + (Half * Z2 + ThirtyTwo * Z2 * Z2)) - One;
    } while ( ! ((Z2 <= X) || (X <= Zero)));
  X = Y = Z = FABS((Three / Four - Two / Three) * Three - One / Four);
  do  {
    Z1 = Z;
    Z = (One / Two - ((One / Two - (Half * Z1 + ThirtyTwo * Z1 * Z1))
      + One / Two)) + One / Two;
    } while ( ! ((Z1 <= Z) || (Z <= Zero)));
  do  {
    do  {
      Y1 = Y;
      Y = (Half - ((Half - (Half * Y1 + ThirtyTwo * Y1 * Y1)) + Half
        )) + Half;
      } while ( ! ((Y1 <= Y) || (Y <= Zero)));
    X1 = X;
    X = ((Half * X1 + ThirtyTwo * X1 * X1) - F9) + F9;
    } while ( ! ((X1 <= X) || (X <= Zero)));
  if ((X1 != Y1) || (X1 != Z1)) {
    BadCond(Serious, "Disagreements among the values X1, Y1, Z1,\n");
    printf("respectively  %.7e,  %.7e,  %.7e,\n", X1, Y1, Z1);
    printf("are symptoms of inconsistencies introduced\n");
    printf("by extra-precise evaluation of arithmetic subexpressions.\n");
    notify("Possibly some part of this");
    if ((X1 == U1) || (Y1 == U1) || (Z1 == U1))  printf(
      "That feature is not tested further by this program.\n") ;
    }
  else  {
    if ((Z1 != U1) || (Z2 != U2)) {
      if ((Z1 >= U1) || (Z2 >= U2)) {
        BadCond(Failure, "");
        notify("Precision");
        printf("\tU1 = %.7e, Z1 - U1 = %.7e\n",U1,Z1-U1);
        printf("\tU2 = %.7e, Z2 - U2 = %.7e\n",U2,Z2-U2);
        }
      else {
        if ((Z1 <= Zero) || (Z2 <= Zero)) {
          printf("Because of unusual Radix = %f", Radix);
          printf(", or exact rational arithmetic a result\n");
          printf("Z1 = %.7e, or Z2 = %.7e ", Z1, Z2);
          notify("of an\nextra-precision");
          }
        if (Z1 != Z2 || Z1 > Zero) {
          X = Z1 / U1;
          Y = Z2 / U2;
          if (Y > X) X = Y;
          Q = - LOG(X);
          printf("Some subexpressions appear to be calculated extra\n");
          printf("precisely with about %g extra B-digits, i.e.\n",
            (Q / LOG(Radix)));
          printf("roughly %g extra significant decimals.\n",
            Q / LOG(10.));
          }
        printf("That feature is not tested further by this program.\n");
        }
      }
    }
  Pause();
  /*=============================================*/
  Milestone = 35;
  /*=============================================*/
  if (Radix >= Two) {
    X = W / (Radix * Radix);
    Y = X + One;
    Z = Y - X;
    T = Z + U2;
    X = T - Z;
    TstCond (Failure, X == U2,
      "Subtraction is not normalized X=Y,X+Z != Y+Z!");
    if (X == U2) printf(
      "Subtraction appears to be normalized, as it should be.");
    }
  printf("\nChecking for guard digit in *, /, and -.\n");
  Y = F9 * One;
  Z = One * F9;
  X = F9 - Half;
  Y = (Y - Half) - X;
  Z = (Z - Half) - X;
  X = One + U2;
  T = X * Radix;
  R = Radix * X;
  X = T - Radix;
  X = X - Radix * U2;
  T = R - Radix;
  T = T - Radix * U2;
  X = X * (Radix - One);
  T = T * (Radix - One);
  if ((X == Zero) && (Y == Zero) && (Z == Zero) && (T == Zero)) GMult = Yes;
  else {
    GMult = No;
    TstCond (Serious, False,
      "* lacks a Guard Digit, so 1*X != X");
    }
  Z = Radix * U2;
  X = One + Z;
  Y = FABS((X + Z) - X * X) - U2;
  X = One - U2;
  Z = FABS((X - U2) - X * X) - U1;
  TstCond (Failure, (Y <= Zero)
       && (Z <= Zero), "* gets too many final digits wrong.\n");
  Y = One - U2;
  X = One + U2;
  Z = One / Y;
  Y = Z - X;
  X = One / Three;
  Z = Three / Nine;
  X = X - Z;
  T = Nine / TwentySeven;
  Z = Z - T;
  TstCond(Defect, X == Zero && Y == Zero && Z == Zero,
    "Division lacks a Guard Digit, so error can exceed 1 ulp\n\
or  1/3  and  3/9  and  9/27 may disagree");
  Y = F9 / One;
  X = F9 - Half;
  Y = (Y - Half) - X;
  X = One + U2;
  T = X / One;
  X = T - X;
  if ((X == Zero) && (Y == Zero) && (Z == Zero)) GDiv = Yes;
  else {
    GDiv = No;
    TstCond (Serious, False,
      "Division lacks a Guard Digit, so X/1 != X");
    }
  X = One / (One + U2);
  Y = X - Half - Half;
  TstCond (Serious, Y < Zero,
       "Computed value of 1/1.000..1 >= 1");
  X = One - U2;
  Y = One + Radix * U2;
  Z = X * Radix;
  T = Y * Radix;
  R = Z / Radix;
  StickyBit = T / Radix;
  X = R - X;
  Y = StickyBit - Y;
  TstCond (Failure, X == Zero && Y == Zero,
      "* and/or / gets too many last digits wrong");
  Y = One - U1;
  X = One - F9;
  Y = One - Y;
  T = Radix - U2;
  Z = Radix - BMinusU2;
  T = Radix - T;
  if ((X == U1) && (Y == U1) && (Z == U2) && (T == U2)) GAddSub = Yes;
  else {
    GAddSub = No;
    TstCond (Serious, False,
      "- lacks Guard Digit, so cancellation is obscured");
    }
  if (F9 != One && F9 - One >= Zero) {
    BadCond(Serious, "comparison alleges  (1-U1) < 1  although\n");
    printf("  subtraction yields  (1-U1) - 1 = 0 , thereby vitiating\n");
    printf("  such precautions against division by zero as\n");
    printf("  ...  if (X == 1.0) {.....} else {.../(X-1.0)...}\n");
    }
  if (GMult == Yes && GDiv == Yes && GAddSub == Yes) printf(
    "     *, /, and - appear to have guard digits, as they should.\n");
/*=============================================*/
  Milestone = 40;
/*=============================================*/
  Pause();
  printf("Checking rounding on multiply, divide and add/subtract.\n");
  RMult = Other;
  RDiv = Other;
  RAddSub = Other;
  RadixD2 = Radix / Two;
  A1 = Two;
  Done = False;
  do  {
    AInvrse = Radix;
    do  {
      X = AInvrse;
      AInvrse = AInvrse / A1;
      } while ( ! (FLOOR(AInvrse) != AInvrse));
    Done = (X == One) || (A1 > Three);
    if (! Done) A1 = Nine + One;
    } while ( ! (Done));
  if (X == One) A1 = Radix;
  AInvrse = One / A1;
  X = A1;
  Y = AInvrse;
  Done = False;
  do  {
    Z = X * Y - Half;
    TstCond (Failure, Z == Half,
      "X * (1/X) differs from 1");
    Done = X == Radix;
    X = Radix;
    Y = One / X;
    } while ( ! (Done));
  Y2 = One + U2;
  Y1 = One - U2;
  X = OneAndHalf - U2;
  Y = OneAndHalf + U2;
  Z = (X - U2) * Y2;
  T = Y * Y1;
  Z = Z - X;
  T = T - X;
  X = X * Y2;
  Y = (Y + U2) * Y1;
  X = X - OneAndHalf;
  Y = Y - OneAndHalf;

  if ((X == Zero) && (Y == Zero) && (Z == Zero) && (T <= Zero))
  {
    X = (OneAndHalf + U2) * Y2;
    Y = OneAndHalf - U2 - U2;
    Z = OneAndHalf + U2 + U2;
    T = (OneAndHalf - U2) * Y1;
    X = X - (Z + U2);
    StickyBit = Y * Y1;
    S = Z * Y2;
    T = T - Y;
    Y = (U2 - Y) + StickyBit;
    Z = S - (Z + U2 + U2);
    StickyBit = (Y2 + U2) * Y1;
    Y1 = Y2 * Y1;
    StickyBit = StickyBit - Y2;
    Y1 = Y1 - Half;

    if ((X == Zero) && (Y == Zero) && (Z == Zero) && (T == Zero)
      && ( StickyBit == Zero) && (Y1 == Half))
    {
      RMult = Rounded;
      printf("Multiplication appears to round correctly.\n");
    }
    else if ((X + U2 == Zero) && (Y < Zero) && (Z + U2 == Zero)
        && (T < Zero) && (StickyBit + U2 == Zero)
        && (Y1 < Half))
    {
        RMult = Chopped;
        printf("Multiplication appears to chop.\n");
    }
    else
    { 
      printf("* is neither chopped nor correctly rounded.\n");
    }

    if ((RMult == Rounded) && (GMult == No)) notify("Multiplication");
    }
  else printf("* is neither chopped nor correctly rounded.\n");
/*=============================================*/
  Milestone = 45;
/*=============================================*/
  Y2 = One + U2;
  Y1 = One - U2;
  Z = OneAndHalf + U2 + U2;
  X = Z / Y2;
  T = OneAndHalf - U2 - U2;
  Y = (T - U2) / Y1;
  Z = (Z + U2) / Y2;
  X = X - OneAndHalf;
  Y = Y - T;
  T = T / Y1;
  Z = Z - (OneAndHalf + U2);
  T = (U2 - OneAndHalf) + T;
  if (! ((X > Zero) || (Y > Zero) || (Z > Zero) || (T > Zero))) {
    X = OneAndHalf / Y2;
    Y = OneAndHalf - U2;
    Z = OneAndHalf + U2;
    X = X - Y;
    T = OneAndHalf / Y1;
    Y = Y / Y1;
    T = T - (Z + U2);
    Y = Y - Z;
    Z = Z / Y2;
    Y1 = (Y2 + U2) / Y2;
    Z = Z - OneAndHalf;
    Y2 = Y1 - Y2;
    Y1 = (F9 - U1) / F9;
    if ((X == Zero) && (Y == Zero) && (Z == Zero) && (T == Zero)
      && (Y2 == Zero) && (Y2 == Zero)
      && (Y1 - Half == F9 - Half )) {
      RDiv = Rounded;
      printf("Division appears to round correctly.\n");
      if (GDiv == No) notify("Division");
      }
    else if ((X < Zero) && (Y < Zero) && (Z < Zero) && (T < Zero)
      && (Y2 < Zero) && (Y1 - Half < F9 - Half)) {
      RDiv = Chopped;
      printf("Division appears to chop.\n");
      }
    }
  if (RDiv == Other) printf("/ is neither chopped nor correctly rounded.\n");
  BInvrse = One / Radix;
  TstCond (Failure, (BInvrse * Radix - Half == Half),
       "Radix * ( 1 / Radix ) differs from 1");
/*=============================================*/
  Milestone = 50;
/*=============================================*/
  TstCond (Failure, ((F9 + U1) - Half == Half)
       && ((BMinusU2 + U2 ) - One == Radix - One),
       "Incomplete carry-propagation in Addition");
  X = One - U1 * U1;
  Y = One + U2 * (One - U2);
  Z = F9 - Half;
  X = (X - Half) - Z;
  Y = Y - One;
  if ((X == Zero) && (Y == Zero)) {
    RAddSub = Chopped;
    printf("Add/Subtract appears to be chopped.\n");
    }
  if (GAddSub == Yes)
  {
    X = (Half + U2) * U2;
    Y = (Half - U2) * U2;
    X = One + X;
    Y = One + Y;
    X = (One + U2) - X;
    Y = One - Y;
    if ((X == Zero) && (Y == Zero))
    {
      X = (Half + U2) * U1;
      Y = (Half - U2) * U1;
      X = One - X;
      Y = One - Y;
      X = F9 - X;
      Y = One - Y;
      if ((X == Zero) && (Y == Zero)) {
        RAddSub = Rounded;
        printf("Addition/Subtraction appears to round correctly.\n");
        if (GAddSub == No) notify("Add/Subtract");
        }
      else printf("Addition/Subtraction neither rounds nor chops.\n");
      }
    else printf("Addition/Subtraction neither rounds nor chops.\n");
    }
  else printf("Addition/Subtraction neither rounds nor chops.\n");
  S = One;
  X = One + Half * (One + Half);
  Y = (One + U2) * Half;
  Z = X - Y;
  T = Y - X;
  StickyBit = Z + T;
  if (StickyBit != Zero) {
    S = Zero;
    BadCond(Flaw, "(X - Y) + (Y - X) is non zero!\n");
    }
  StickyBit = Zero;
  if ((GMult == Yes) && (GDiv == Yes) && (GAddSub == Yes)
    && (RMult == Rounded) && (RDiv == Rounded)
    && (RAddSub == Rounded) && (FLOOR(RadixD2) == RadixD2)) {
    printf("Checking for sticky bit.\n");
    X = (Half + U1) * U2;
    Y = Half * U2;
    Z = One + Y;
    T = One + X;
    if ((Z - One <= Zero) && (T - One >= U2)) {
      Z = T + Y;
      Y = Z - X;
      if ((Z - T >= U2) && (Y - T == Zero)) {
        X = (Half + U1) * U1;
        Y = Half * U1;
        Z = One - Y;
        T = One - X;
        if ((Z - One == Zero) && (T - F9 == Zero)) {
          Z = (Half - U1) * U1;
          T = F9 - Z;
          Q = F9 - Y;
          if ((T - F9 == Zero) && (F9 - U1 - Q == Zero)) {
            Z = (One + U2) * OneAndHalf;
            T = (OneAndHalf + U2) - Z + U2;
            X = One + Half / Radix;
            Y = One + Radix * U2;
            Z = X * Y;
            if (T == Zero && X + Radix * U2 - Z == Zero) {
              if (Radix != Two) {
                X = Two + U2;
                Y = X / Two;
                if ((Y - One == Zero)) StickyBit = S;
                }
              else StickyBit = S;
              }
            }
          }
        }
      }
    }

  if (StickyBit == One) 
  {
    printf("Sticky bit apparently used correctly.\n");
  }
  else
  { 
    printf("Sticky bit used incorrectly or not at all.\n");
  }

  TstCond (Flaw, !(GMult == No || GDiv == No || GAddSub == No ||
      RMult == Other || RDiv == Other || RAddSub == Other),
    "lack(s) of guard digits or failure(s) to correctly round or chop\n\
(noted above) count as one flaw in the final tally below");
/*=============================================*/
  Milestone = 60;
/*=============================================*/
  printf("\n");
  printf("Does Multiplication commute?  ");
  printf("Testing on %d random pairs.\n", NoTrials);
  Random9 = SQRT(3.0);
  Random1 = Third;
  I = 1;
  do  {
    X = Random();
    Y = Random();
    Z9 = Y * X;
    Z = X * Y;
    Z9 = Z - Z9;
    I = I + 1;
    } while ( ! ((I > NoTrials) || (Z9 != Zero)));
  if (I == NoTrials)
  {
    Random1 = One + Half / Three;
    Random2 = (U2 + U1) + One;
    Z = Random1 * Random2;
    Y = Random2 * Random1;
    Z9 = (One + Half / Three) * ((U2 + U1) + One) - (One + Half /
      Three) * ((U2 + U1) + One);
    }
  if (! ((I == NoTrials) || (Z9 == Zero)))
    BadCond(Defect, "X * Y == Y * X trial fails.\n");
  else printf("     No failures found in %d integer pairs.\n", NoTrials);
/*=============================================*/
  Milestone = 70;
/*=============================================*/
  printf("\nRunning test of square root(x).\n");
  TstCond (Failure, (Zero == SQRT(Zero))
       && (- Zero == SQRT(- Zero))
       && (One == SQRT(One)), "Square root of 0.0, -0.0 or 1.0 wrong");
  MinSqEr = Zero;
  MaxSqEr = Zero;
  J = Zero;
  X = Radix;
  OneUlp = U2;
  SqXMinX (Serious);
  X = BInvrse;
  OneUlp = BInvrse * U1;
  SqXMinX (Serious);
  X = U1;
  OneUlp = U1 * U1;
  SqXMinX (Serious);
  if (J != Zero) Pause();
  printf("Testing if sqrt(X * X) == X for %d Integers X.\n", NoTrials);
  J = Zero;
  X = Two;
  Y = Radix;
  if ((Radix != One)) do  {
    X = Y;
    Y = Radix * Y;
    } while ( ! ((Y - X >= NoTrials)));
  OneUlp = X * U2;
  I = 1;
  while (I <= NoTrials) {
    X = X + One;
    SqXMinX (Defect);
    if (J > Zero) break;
    I = I + 1;
    }
  printf("Test for sqrt monotonicity.\n");
  I = - 1;
  X = BMinusU2;
  Y = Radix;
  Z = Radix + Radix * U2;
  NotMonot = False;
  Monot = False;
  while ( ! (NotMonot || Monot)) {
    I = I + 1;
    X = SQRT(X);
    Q = SQRT(Y);
    Z = SQRT(Z);
    if ((X > Q) || (Q > Z)) NotMonot = True;
    else {
      Q = FLOOR(Q + Half);
      if (!(I > 0 || Radix == Q * Q)) Monot = True;
      else if (I > 0) {
      if (I > 1) Monot = True;
      else {
        Y = Y * BInvrse;
        X = Y - U1;
        Z = Y + U1;
        }
      }
      else {
        Y = Q;
        X = Y - U2;
        Z = Y + U2;
        }
      }
    }
  if (Monot) printf("sqrt has passed a test for Monotonicity.\n");
  else {
    BadCond(Defect, "");
    printf("sqrt(X) is non-monotonic for X near %.7e .\n", Y);
    }
/*=============================================*/
  Milestone = 80;
/*=============================================*/
  MinSqEr = MinSqEr + Half;
  MaxSqEr = MaxSqEr - Half;
  Y = (SQRT(One + U2) - One) / U2;
  SqEr = (Y - One) + U2 / Eight;
  if (SqEr > MaxSqEr) MaxSqEr = SqEr;
  SqEr = Y + U2 / Eight;
  if (SqEr < MinSqEr) MinSqEr = SqEr;
  Y = ((SQRT(F9) - U2) - (One - U2)) / U1;
  SqEr = Y + U1 / Eight;
  if (SqEr > MaxSqEr) MaxSqEr = SqEr;
  SqEr = (Y + One) + U1 / Eight;
  if (SqEr < MinSqEr) MinSqEr = SqEr;
  OneUlp = U2;
  X = OneUlp;
  for( Indx = 1; Indx <= 3; ++Indx) {
    Y = SQRT((X + U1 + X) + F9);
    Y = ((Y - U2) - ((One - U2) + X)) / OneUlp;
    Z = ((U1 - X) + F9) * Half * X * X / OneUlp;
    SqEr = (Y + Half) + Z;
    if (SqEr < MinSqEr) MinSqEr = SqEr;
    SqEr = (Y - Half) + Z;
    if (SqEr > MaxSqEr) MaxSqEr = SqEr;
    if (((Indx == 1) || (Indx == 3))) 
      X = OneUlp * Sign (X) * FLOOR(Eight / (Nine * SQRT(OneUlp)));
    else {
      OneUlp = U1;
      X = - OneUlp;
      }
    }
/*=============================================*/
  Milestone = 85;
/*=============================================*/
  SqRWrng = False;
  Anomaly = False;
  RSqrt = Other; /* ~dgh */
  if (Radix != One) {
    printf("Testing whether sqrt is rounded or chopped.\n");
    D = FLOOR(Half + POW(Radix, One + Precision - FLOOR(Precision)));
/* ... == Radix^(1 + fract) if (Precision == Integer + fract. */
    X = D / Radix;
    Y = D / A1;
    if ((X != FLOOR(X)) || (Y != FLOOR(Y))) {
      Anomaly = True;
      }
    else {
      X = Zero;
      Z2 = X;
      Y = One;
      Y2 = Y;
      Z1 = Radix - One;
      FourD = Four * D;
      do  {
        if (Y2 > Z2) {
          Q = Radix;
          Y1 = Y;
          do  {
            X1 = FABS(Q + FLOOR(Half - Q / Y1) * Y1);
            Q = Y1;
            Y1 = X1;
            } while ( ! (X1 <= Zero));
          if (Q <= One) {
            Z2 = Y2;
            Z = Y;
            }
          }
        Y = Y + Two;
        X = X + Eight;
        Y2 = Y2 + X;
        if (Y2 >= FourD) Y2 = Y2 - FourD;
        } while ( ! (Y >= D));
      X8 = FourD - Z2;
      Q = (X8 + Z * Z) / FourD;
      X8 = X8 / Eight;
      if (Q != FLOOR(Q)) Anomaly = True;
      else {
        Break = False;
        do  {
          X = Z1 * Z;
          X = X - FLOOR(X / Radix) * Radix;
          if (X == One) 
            Break = True;
          else
            Z1 = Z1 - One;
          } while ( ! (Break || (Z1 <= Zero)));
        if ((Z1 <= Zero) && (! Break)) Anomaly = True;
        else {
          if (Z1 > RadixD2) Z1 = Z1 - Radix;
          do  {
            NewD();
            } while ( ! (U2 * D >= F9));
          if (D * Radix - D != W - D) Anomaly = True;
          else {
            Z2 = D;
            I = 0;
            Y = D + (One + Z) * Half;
            X = D + Z + Q;
            SR3750();
            Y = D + (One - Z) * Half + D;
            X = D - Z + D;
            X = X + Q + X;
            SR3750();
            NewD();
            if (D - Z2 != W - Z2) Anomaly = True;
            else {
              Y = (D - Z2) + (Z2 + (One - Z) * Half);
              X = (D - Z2) + (Z2 - Z + Q);
              SR3750();
              Y = (One + Z) * Half;
              X = Q;
              SR3750();
              if (I == 0) Anomaly = True;
              }
            }
          }
        }
      }
    if ((I == 0) || Anomaly) {
      BadCond(Failure, "Anomalous arithmetic with Integer < ");
      printf("Radix^Precision = %.7e\n", W);
      printf(" fails test whether sqrt rounds or chops.\n");
      SqRWrng = True;
      }
    }
  if (! Anomaly) {
    if (! ((MinSqEr < Zero) || (MaxSqEr > Zero))) {
      RSqrt = Rounded;
      printf("Square root appears to be correctly rounded.\n");
      }
    else  {
      if ((MaxSqEr + U2 > U2 - Half) || (MinSqEr > Half)
        || (MinSqEr + Radix < Half)) SqRWrng = True;
      else {
        RSqrt = Chopped;
        printf("Square root appears to be chopped.\n");
        }
      }
    }
  if (SqRWrng) {
    printf("Square root is neither chopped nor correctly rounded.\n");
    printf("Observed errors run from %.7e ", MinSqEr - Half);
    printf("to %.7e ulps.\n", Half + MaxSqEr);
    TstCond (Serious, MaxSqEr - MinSqEr < Radix * Radix,
      "sqrt gets too many last digits wrong");
    }
/*=============================================*/
  Milestone = 90;
/*=============================================*/
  Pause();
  printf("Testing powers Z^i for small Integers Z and i.\n");
  N = 0;
/* ... test powers of zero. */
  I = 0;
  Z = -Zero;
  M = 3;
  Break = False;
  do  {
    X = One;
    SR3980();
    if (I <= 10) {
      I = 1023;
      SR3980();
      }
    if (Z == MinusOne) Break = True;
    else {
      Z = MinusOne;
/* .. if(-1)^N is invalid, replace MinusOne by One. */
      I = - 4;
      }
    } while ( ! Break);
  PrintIfNPositive();
  N1 = N;
  N = 0;
  Z = A1;
  M = (int)FLOOR(Two * LOG(W) / LOG(A1));
  Break = False;
  do  {
    X = Z;
    I = 1;
    SR3980();
    if (Z == AInvrse) Break = True;
    else Z = AInvrse;
    } while ( ! (Break));
  /*=============================================*/
    Milestone = 100;
  /*=============================================*/
  /*  Powers of Radix have been tested, */
  /*         next try a few primes     */
  M = NoTrials;
  Z = Three;
  do  {
    X = Z;
    I = 1;
    SR3980();
    do  {
      Z = Z + Two;
      } while ( Three * FLOOR(Z / Three) == Z );
    } while ( Z < Eight * Three );
  if (N > 0) {
    printf("Errors like this may invalidate financial calculations\n");
    printf("\tinvolving interest rates.\n");
    }
  PrintIfNPositive();
  N += N1;
  if (N == 0) printf("... no discrepancies found.\n");
  if (N > 0) Pause();
  else printf("\n");
  /*=============================================*/
  Milestone = 110;
  /*=============================================*/
  printf("Seeking Underflow thresholds UfThold and E0.\n");
  D = U1;
  if (Precision != FLOOR(Precision)) {
    D = BInvrse;
    X = Precision;
    do  {
      D = D * BInvrse;
      X = X - One;
      } while ( X > Zero);
    }
  Y = One;
  Z = D;
  /* ... D is power of 1/Radix < 1. */
  do  {
    C = Y;
    Y = Z;
    Z = Y * Y;
    } while ((Y > Z) && (Z + Z > Z));
  Y = C;
  Z = Y * D;
  do  {
    C = Y;
    Y = Z;
    Z = Y * D;
    } while ((Y > Z) && (Z + Z > Z));
  if (Radix < Two) HInvrse = Two;
  else HInvrse = Radix;
  H = One / HInvrse;
/* ... 1/HInvrse == H == Min(1/Radix, 1/2) */
  CInvrse = One / C;
  E0 = C;
  Z = E0 * H;
/* ...1/Radix^(BIG Integer) << 1 << CInvrse == 1/C */
  do  {
    Y = E0;
    E0 = Z;
    Z = E0 * H;
    } while ((E0 > Z) && (Z + Z > Z));
  UfThold = E0;
  E1 = Zero;
  Q = Zero;
  E9 = U2;
  S = One + E9;
  D = C * S;
  if (D <= C) {
    E9 = Radix * U2;
    S = One + E9;
    D = C * S;
    if (D <= C) {
      BadCond(Failure, "multiplication gets too many last digits wrong.\n");
      Underflow = E0;
      Y1 = Zero;
      PseudoZero = Z;
      Pause();
      }
    }
  else {
    Underflow = D;
    PseudoZero = Underflow * H;
    UfThold = Zero;
    do  {
      Y1 = Underflow;
      Underflow = PseudoZero;
      if (E1 + E1 <= E1) {
        Y2 = Underflow * HInvrse;
        E1 = FABS(Y1 - Y2);
        Q = Y1;
        if ((UfThold == Zero) && (Y1 != Y2)) UfThold = Y1;
        }
      PseudoZero = PseudoZero * H;
      } while ((Underflow > PseudoZero)
        && (PseudoZero + PseudoZero > PseudoZero));
    }
/* Comment line 4530 .. 4560 */
  if (PseudoZero != Zero) {
    printf("\n");
    Z = PseudoZero;
/* ... Test PseudoZero for "phoney- zero" violates */
/* ... PseudoZero < Underflow or PseudoZero < PseudoZero + PseudoZero
       ... */
    if (PseudoZero <= Zero) {
      BadCond(Failure, "Positive expressions can underflow to an\n");
      printf("allegedly negative value\n");
      printf("PseudoZero that prints out as: %g .\n", PseudoZero);
      X = - PseudoZero;
      if (X <= Zero) {
        printf("But -PseudoZero, which should be\n");
        printf("positive, isn't; it prints out as  %g .\n", X);
        }
      }
    else {
      BadCond(Flaw, "Underflow can stick at an allegedly positive\n");
      printf("value PseudoZero that prints out as %g .\n", PseudoZero);
      }
    TstPtUf();
    }
/*=============================================*/
  Milestone = 120;
/*=============================================*/

  if ( CInvrse * Y > CInvrse * Y1 )
  {
    S = H * S;
    E0 = Underflow;
  }

  if ( ! ((E1 == Zero) || (E1 == E0)) )
  {
    BadCond(Defect, "");
    if (E1 < E0)
    {
      printf("Products underflow at a higher");
      printf(" threshold than differences.\n");
      if (PseudoZero == Zero) 
      E0 = E1;
    }
    else
    {
      printf("Difference underflows at a higher");
      printf(" threshold than products.\n");
    }
  }

  printf("Smallest strictly positive number found is E0 = %g .\n", E0);
  Z = E0;
  TstPtUf();
  Underflow = E0;
  if (N == 1) Underflow = Y;
  I = 4;
  if (E1 == Zero) I = 3;
  if (UfThold == Zero) I = I - 2;
  UfNGrad = True;
  switch (I)  {
    case  1:
    UfThold = Underflow;
    if ((CInvrse * Q) != ((CInvrse * Y) * S)) {
      UfThold = Y;
      BadCond(Failure, "Either accuracy deteriorates as numbers\n");
      printf("approach a threshold = %.17e\n", UfThold);;
      printf(" coming down from %.17e\n", C);
      printf(" or else multiplication gets too many last digits wrong.\n");
      }
    Pause();
    break;
  
    case  2:
    BadCond(Failure, "Underflow confuses Comparison, which alleges that\n");
    printf("Q == Y while denying that |Q - Y| == 0; these values\n");
    printf("print out as Q = %.17e, Y = %.17e .\n", Q, Y2);
    printf ("|Q - Y| = %.17e .\n" , FABS(Q - Y2));
    UfThold = Q;
    break;
  
    case  3:
    X = X;
    break;
  
    case  4:
    if ((Q == UfThold) && (E1 == E0)
      && (FABS( UfThold - E1 / E9) <= E1)) {
      UfNGrad = False;
      printf("Underflow is gradual; it incurs Absolute Error =\n");
      printf("(roundoff in UfThold) < E0.\n");
      Y = E0 * CInvrse;
      Y = Y * (OneAndHalf + U2);
      X = CInvrse * (One + U2);
      Y = Y / X;
      IEEE = (Y == E0);
      }
    }
  if (UfNGrad) {
    printf("\n");
    sigsave = sigfpe;
    if (setjmp(ovfl_buf)) {
      printf("Underflow / UfThold failed!\n");
      R = H + H;
      }
    else R = SQRT(Underflow / UfThold);
    sigsave = 0;
    if (R <= H) {
      Z = R * UfThold;
      X = Z * (One + R * H * (One + H));
      }
    else {
      Z = UfThold;
      X = Z * (One + H * H * (One + H));
      }
    if (! ((X == Z) || (X - Z != Zero))) {
      BadCond(Flaw, "");
      printf("X = %.17e\n\tis not equal to Z = %.17e .\n", X, Z);
      Z9 = X - Z;
      printf("yet X - Z yields %.17e .\n", Z9);
      printf("    Should this NOT signal Underflow, ");
      printf("this is a SERIOUS DEFECT\nthat causes ");
      printf("confusion when innocent statements like\n");;
      printf("    if (X == Z)  ...  else");
      printf("  ... (f(X) - f(Z)) / (X - Z) ...\n");
      printf("encounter Division by Zero although actually\n");
      sigsave = sigfpe;
      if (setjmp(ovfl_buf)) printf("X / Z fails!\n");
      else printf("X / Z = 1 + %g .\n", (X / Z - Half) - Half);
      sigsave = 0;
      }
    }
  printf("The Underflow threshold is %.17e, %s\n", UfThold,
       " below which");
  printf("calculation may suffer larger Relative error than ");
  printf("merely roundoff.\n");
  Y2 = U1 * U1;
  Y = Y2 * Y2;
  Y2 = Y * U1;
  if (Y2 <= UfThold) {
    if (Y > E0) {
      BadCond(Defect, "");
      I = 5;
      }
    else {
      BadCond(Serious, "");
      I = 4;
      }
    printf("Range is too narrow; U1^%d Underflows.\n", I);
    }
  /*=============================================*/
  Milestone = 130;
  /*=============================================*/
  Y = - FLOOR(Half - TwoForty * LOG(UfThold) / LOG(HInvrse)) / TwoForty;
  Y2 = Y + Y;
  printf("Since underflow occurs below the threshold\n");
  printf("UfThold = (%.17e) ^ (%.17e)\nonly underflow ", HInvrse, Y);
  printf("should afflict the expression\n\t(%.17e) ^ (%.17e);\n",
    HInvrse, Y2);
  printf("actually calculating yields:");
  if (setjmp(ovfl_buf)) {
    sigsave = 0;
    BadCond(Serious, "trap on underflow.\n");
    }
  else {
    sigsave = sigfpe;
    V9 = POW(HInvrse, Y2);
    sigsave = 0;
    printf(" %.17e .\n", V9);
    if (! ((V9 >= Zero) && (V9 <= (Radix + Radix + E9) * UfThold))) {
      BadCond(Serious, "this is not between 0 and underflow\n");
    printf("   threshold = %.17e .\n", UfThold);
    }
    else if (! (V9 > UfThold * (One + E9)))
      printf("This computed value is O.K.\n");
    else {
      BadCond(Defect, "this is not between 0 and underflow\n");
      printf("   threshold = %.17e .\n", UfThold);
      }
    }
  /*=============================================*/
  Milestone = 140;
  /*=============================================*/
  printf("\n");
  /* ...calculate Exp2 == exp(2) == 7.389056099... */
  X = Zero;
  I = 2;
  Y = Two * Three;
  Q = Zero;
  N = 0;
  do  {
    Z = X;
    I = I + 1;
    Y = Y / (I + I);
    R = Y + Q;
    X = Z + R;
    Q = (Z - X) + R;
    } while(X > Z);
  Z = (OneAndHalf + One / Eight) + X / (OneAndHalf * ThirtyTwo);
  X = Z * Z;
  Exp2 = X * X;
  X = F9;
  Y = X - U1;
  printf("Testing X^((X + 1) / (X - 1)) vs. exp(2) = %.17e as X -> 1.\n",
    Exp2);
  for(I = 1;;) {
    Z = X - BInvrse;
    Z = (X + One) / (Z - (One - BInvrse));
    Q = POW(X, Z) - Exp2;
    if (FABS(Q) > TwoForty * U2) {
      N = 1;
       V9 = (X - BInvrse) - (One - BInvrse);
      BadCond(Defect, "Calculated");
      printf(" %.17e for\n", POW(X,Z));
      printf("\t(1 + (%.17e) ^ (%.17e);\n", V9, Z);
      printf("\tdiffers from correct value by %.17e .\n", Q);
      printf("\tThis much error may spoil financial\n");
      printf("\tcalculations involving tiny interest rates.\n");
      break;
      }
    else {
      Z = (Y - X) * Two + Y;
      X = Y;
      Y = Z;
      Z = One + (X - F9)*(X - F9);
      if (Z > One && I < NoTrials) I++;
      else  {
        if (X > One) {
          if (N == 0)
             printf("Accuracy seems adequate.\n");
          break;
          }
        else {
          X = One + U2;
          Y = U2 + U2;
          Y += X;
          I = 1;
          }
        }
      }
    }
  /*=============================================*/
  Milestone = 150;
  /*=============================================*/
  printf("Testing powers Z^Q at four nearly extreme values.\n");
  N = 0;
  Z = A1;
  Q = FLOOR(Half - LOG(C) / LOG(A1));
  Break = False;
  do  {
    X = CInvrse;
    Y = POW(Z, Q);
    IsYeqX();
    Q = - Q;
    X = C;
    Y = POW(Z, Q);
    IsYeqX();
    if (Z < One) Break = True;
    else Z = AInvrse;
    } while ( ! (Break));
  PrintIfNPositive();
  if (N == 0) printf(" ... no discrepancies found.\n");
  printf("\n");
  
  /*=============================================*/
  Milestone = 160;
  /*=============================================*/
  Pause();
  printf("Searching for Overflow threshold:\n");
  printf("This may generate an error.\n");
  Y = - CInvrse;
  V9 = HInvrse * Y;
  sigsave = sigfpe;
  if (setjmp(ovfl_buf)) { I = 0; V9 = Y; goto overflow; }
  do {
    V = Y;
    Y = V9;
    V9 = HInvrse * Y;
    } while(V9 < Y);
  I = 1;
overflow:
  sigsave = 0;
  Z = V9;
  printf("Can `Z = -Y' overflow?\n");
  printf("Trying it on Y = %.17e .\n", Y);
  V9 = - Y;
  V0 = V9;

  if (V - Y == V + V0)
  {
    printf("Seems O.K.\n");
  }
  else
  {
    printf("finds a ");
    BadCond(Flaw, "-(-Y) differs from Y.\n");
  }

  if (Z != Y)
  {
    BadCond(Serious, "");
    printf("overflow past %.17e\n\tshrinks to %.17e .\n", Y, Z);
    }
  if (I)
  {
    Y = V * (HInvrse * U2 - HInvrse);
    Z = Y + ((One - HInvrse) * U2) * V;
    if (Z < V0) Y = Z;
    if (Y < V0) V = Y;
    if (V0 - V < V0) V = V0;
    }
  else {
    V = Y * (HInvrse * U2 - HInvrse);
    V = V + ((One - HInvrse) * U2) * Y;
    }
  printf("Overflow threshold is V  = %.17e .\n", V);
  if (I) printf("Overflow saturates at V0 = %.17e .\n", V0);
  else printf("There is no saturation value because \
the system traps on overflow.\n");
  V9 = V * One;
  printf("No Overflow should be signaled for V * 1 = %.17e\n", V9);
  V9 = V / One;
  printf("                           nor for V / 1 = %.17e .\n", V9);
  printf("Any overflow signal separating this * from the one\n");
  printf("above is a DEFECT.\n");
  /*=============================================*/
  Milestone = 170;
  /*=============================================*/
  if (!(-V < V && -V0 < V0 && -UfThold < V && UfThold < V)) {
    BadCond(Failure, "Comparisons involving ");
    printf("+-%g, +-%g\nand +-%g are confused by Overflow.",
      V, V0, UfThold);
    }
  /*=============================================*/
  Milestone = 175;
  /*=============================================*/
  printf("\n");
  for(Indx = 1; Indx <= 3; ++Indx) {
    switch (Indx)  {
      case 1: Z = UfThold; break;
      case 2: Z = E0; break;
      case 3: Z = PseudoZero; break;
      }
    if (Z != Zero) {
      V9 = SQRT(Z);
      Y = V9 * V9;
      if (Y / (One - Radix * E9) < Z
         || Y > (One + Radix * E9) * Z) { /* dgh: + E9 --> * E9 */
        if (V9 > U1) BadCond(Serious, "");
        else BadCond(Defect, "");
        printf("Comparison alleges that what prints as Z = %.17e\n", Z);
        printf(" is too far from sqrt(Z) ^ 2 = %.17e .\n", Y);
        }
      }
    }
  /*=============================================*/
  Milestone = 180;
  /*=============================================*/
  for(Indx = 1; Indx <= 2; ++Indx) {
    if (Indx == 1) Z = V;
    else Z = V0;
    V9 = SQRT(Z);
    X = (One - Radix * E9) * V9;
    V9 = V9 * X;
    if (((V9 < (One - Two * Radix * E9) * Z) || (V9 > Z)))
    {
      Y = V9;
      if (X < W) BadCond(Serious, "");
      else BadCond(Defect, "");
      printf("Comparison alleges that Z = %17e\n", Z);
      printf(" is too far from sqrt(Z) ^ 2 (%.17e) .\n", Y);
      }
    }
  /*=============================================*/
  Milestone = 190;
  /*=============================================*/
  Pause();
  X = UfThold * V;
  Y = Radix * Radix;
  if (X*Y < One || X > Y)
  {
    if (X * Y < U1 || X > Y/U1) BadCond(Defect, "Badly");
    else BadCond(Flaw, "");
      
    printf(" unbalanced range; UfThold * V = %.17e\n\t%s\n",
      X, "is too far from 1.\n");
    }
  /*=============================================*/
  Milestone = 200;
  /*=============================================*/
  for (Indx = 1; Indx <= 5; ++Indx)  {
    X = F9;
    switch (Indx)  {
      case 2: X = One + U2; break;
      case 3: X = V; break;
      case 4: X = UfThold; break;
      case 5: X = Radix;
      }
    Y = X;
    sigsave = sigfpe;
    if (setjmp(ovfl_buf))
      printf("  X / X  traps when X = %g\n", X);
    else {
      V9 = (Y / X - Half) - Half;
      if (V9 == Zero) continue;
      if (V9 == - U1 && Indx < 5) BadCond(Flaw, "");
      else BadCond(Serious, "");
      printf("  X / X differs from 1 when X = %.17e\n", X);
      printf("  instead, X / X - 1/2 - 1/2 = %.17e .\n", V9);
      }
    sigsave = 0;
    }
  /*=============================================*/
  Milestone = 210;
  /*=============================================*/
  MyZero = Zero;
  printf("\n");
  printf("What message and/or values does Division by Zero produce?\n") ;
#ifndef NOPAUSE
  printf("This can interupt your program.  You can ");
  printf("skip this part if you wish.\n");
  printf("Do you wish to compute 1 / 0? ");
  fflush(stdout);
  read (KEYBOARD, ch, 8);
  if ((ch[0] == 'Y') || (ch[0] == 'y')) {
#endif
    sigsave = sigfpe;
    printf("    Trying to compute 1 / 0 produces ...");
    if (!setjmp(ovfl_buf)) printf("  %.7e .\n", One / MyZero);
    sigsave = 0;
#ifndef NOPAUSE
    }
  else printf("O.K.\n");
  printf("\nDo you wish to compute 0 / 0? ");
  fflush(stdout);
  read (KEYBOARD, ch, 80);
  if ((ch[0] == 'Y') || (ch[0] == 'y')) {
#endif
    sigsave = sigfpe;
    printf("\n    Trying to compute 0 / 0 produces ...");
    if (!setjmp(ovfl_buf)) printf("  %.7e .\n", Zero / MyZero);
    sigsave = 0;
#ifndef NOPAUSE
    }
  else printf("O.K.\n");
#endif

/*=============================================*/
  Milestone = 220;
/*=============================================*/

  Pause();
  printf("\n");
  {
    static char *msg[] = {
      "FAILUREs  encountered =",
      "SERIOUS DEFECTs  discovered =",
      "DEFECTs  discovered =",
      "FLAWs  discovered =" };
    int i;
    for(i = 0; i < 4; i++) if (ErrCnt[i])
      printf("The number of  %-29s %d.\n",
        msg[i], ErrCnt[i]);
    }
  printf("\n");
  if ((ErrCnt[Failure] + ErrCnt[Serious] + ErrCnt[Defect]
      + ErrCnt[Flaw]) > 0) {
    if ((ErrCnt[Failure] + ErrCnt[Serious] + ErrCnt[
      Defect] == 0) && (ErrCnt[Flaw] > 0)) {
      printf("The arithmetic diagnosed seems ");
      printf("Satisfactory though flawed.\n");
      }
    if ((ErrCnt[Failure] + ErrCnt[Serious] == 0)
      && ( ErrCnt[Defect] > 0)) {
      printf("The arithmetic diagnosed may be Acceptable\n");
      printf("despite inconvenient Defects.\n");
      }
    if ((ErrCnt[Failure] + ErrCnt[Serious]) > 0) {
      printf("The arithmetic diagnosed has ");
      printf("unacceptable Serious Defects.\n");
      }
    if (ErrCnt[Failure] > 0) {
      printf("Potentially fatal FAILURE may have spoiled this");
      printf(" program's subsequent diagnoses.\n");
      }
    }
  else
  {
    printf("No failures, defects nor flaws have been discovered.\n");
    if (! ((RMult == Rounded) && (RDiv == Rounded)
      && (RAddSub == Rounded) && (RSqrt == Rounded))) 
      printf("The arithmetic diagnosed seems Satisfactory.\n");
    else {
      if (StickyBit >= One &&
        (Radix - Two) * (Radix - Nine - One) == Zero) {
        printf("Rounding appears to conform to ");
        printf("the proposed IEEE standard P");
        if ((Radix == Two) &&
           ((Precision - Four * Three * Two) *
            ( Precision - TwentySeven -
             TwentySeven + One) == Zero)) 
          printf("754");
        else printf("854");
        if (IEEE)
        {
          printf(".\n");
        }
        else
        {
          printf(",\nexcept for possibly Double Rounding");
          printf(" during Gradual Underflow.\n");
        }
      }
      printf("The arithmetic diagnosed appears to be Excellent!\n");
    }
  }

  if ( fpecount )
  {
    printf("\nA total of %d floating point exceptions were registered.\n",
      fpecount);
  }

  printf ( "\n" );
  printf ( "PARANOIA:\n" );
  printf ( "  Normal end of execution.\n" );

  return 0;
}
/**********************************************************************/

void sigfpe ( int x )

/**********************************************************************/
/* 
  Purpose:

    SIGFPE is the floating point exception receiver.
*/
{
  fpecount++;
  printf("\n* * * FLOATING-POINT ERROR %d * * *\n", x);
  fflush(stdout);
  if (sigsave)
  {
#ifndef NOSIGNAL
    signal(SIGFPE, sigsave);
#endif
    sigsave = 0;
    longjmp(ovfl_buf, 1);
  }
  exit(1);
}
/**********************************************************************/

FLOAT Sign ( FLOAT X )

/**********************************************************************/
/*
  Purpose:

    SIGN returns the sign of a floating point value.
*/
{ 
  return X >= 0. ? 1.0 : -1.0; 
}
/**********************************************************************/

void Pause ( void )

/**********************************************************************/
/*
  Purpose:

    PAUSE pauses program execution until the user hits RETURN.
*/
{
#ifndef NOPAUSE
  char ch[8];

  printf("\nTo continue, press RETURN");
  fflush(stdout);
  read(KEYBOARD, ch, 8);
#endif
  printf("\nDiagnosis resumes after milestone Number %d", Milestone);
  printf("          Page: %d\n\n", PageNo);
  ++Milestone;
  ++PageNo;

  return;
}
/**********************************************************************/

void TstCond ( int K, int Valid, char * T)

/**********************************************************************/
/*
  Purpose:

    TSTCOND checks whether an error has occurred.
*/
{ 
  if (! Valid) 
  { 
    BadCond ( K, T ); 
    printf ( ".\n" ); 
  } 
  return;
}
/**********************************************************************/

void BadCond ( int K, char * T )

/**********************************************************************/
/*
  Purpose:

    BADCOND tracks and prints the various levels of error.
*/
{
  static char *msg[] = { "FAILURE", "SERIOUS DEFECT", "DEFECT", "FLAW" };

  ErrCnt [K] = ErrCnt [K] + 1;

  printf("%s:  %s", msg[K], T);

  return;
}
/**********************************************************************/

FLOAT Random ( void )

/**********************************************************************/
/*  
  Purpose:

    RANDOM computes a new random value.

  Discussion:

    Random computes
     X = (Random1 + Random9)^5
     Random1 = X - FLOOR(X) + 0.000005 * X;
   and returns the new value of Random1
*/
{
  FLOAT X, Y;
  
  X = Random1 + Random9;
  Y = X * X;
  Y = Y * Y;
  X = X * Y;
  Y = X - FLOOR(X);
  Random1 = Y + X * 0.000005;

  return ( Random1 );
}
/**********************************************************************/

void SqXMinX ( int ErrKind)

/**********************************************************************/
/*
  Purpose:

    SQXMINX ???.
*/
{
  FLOAT XA, XB;
  
  XB = X * BInvrse;
  XA = X - XB;
  SqEr = ((SQRT(X * X) - XB) - XA) / OneUlp;
  if (SqEr != Zero)
  {
    if (SqEr < MinSqEr) MinSqEr = SqEr;
    if (SqEr > MaxSqEr) MaxSqEr = SqEr;
    J = J + 1.0;
    BadCond(ErrKind, "\n");
    printf("sqrt( %.17e) - %.17e  = %.17e\n", X * X, X, OneUlp * SqEr);
    printf("\tinstead of correct value 0 .\n");
  }
  return;
}
/**********************************************************************/

void NewD ( void )

/**********************************************************************/
/*
  Purpose:

    NEWD ???.
*/
{
  X = Z1 * Q;
  X = FLOOR(Half - X / Radix) * Radix + X;
  Q = (Q - X * Z) / Radix + X * X * (D / Radix);
  Z = Z - Two * X * D;
  if (Z <= Zero)
  {
    Z = - Z;
    Z1 = - Z1;
  }
  D = Radix * D;
  return;
}
/**********************************************************************/

void SR3750 ( void )

/**********************************************************************/
/*
  Purpose:

    SR3750 ???.
*/
{
  if (! ((X - Radix < Z2 - Radix) || (X - Z2 > W - Z2)))
  {
    I = I + 1;
    X2 = SQRT(X * D);
    Y2 = (X2 - Z2) - (Y - Z2);
    X2 = X8 / (Y - Half);
    X2 = X2 - Half * X2 * X2;
    SqEr = (Y2 + Half) + (Half - X2);
    if (SqEr < MinSqEr) MinSqEr = SqEr;
    SqEr = Y2 - X2;

    if (SqEr > MaxSqEr) 
    {
      MaxSqEr = SqEr;
    }
  }
  return;
}
/**********************************************************************/

void IsYeqX ( void )

/**********************************************************************/
/*
  Purpose:

    ISYEQX checks whether two quantities are equal, as they should be.
*/
{
  if (Y != X)
  {
    if (N <= 0)
    {

      if (Z == Zero && Q <= Zero)
        printf ( "WARNING:  computing\n" );
      else 
        BadCond(Defect, "computing\n");

      printf ( "\t(%.17e) ^ (%.17e)\n", Z, Q);
      printf ( "\tyielded %.17e;\n", Y);
      printf ( "\twhich compared unequal to correct %.17e ;\n",
        X);
      printf ( "\t\tthey differ by %.17e .\n", Y - X);
    }
/*
  Count the discrepancies.
*/
    N = N + 1;
  }
  return;
}
/**********************************************************************/

void SR3980 ( void )

/**********************************************************************/
/*
  Purpose:

    SR3980 ???
*/
{
  do
  {
    Q = (FLOAT) I;
    Y = POW(Z, Q);
    IsYeqX();
    if ( ++I > M ) 
    {
      break;
    }
    X = Z * X;
  } while ( X < W );

  return;
}
/**********************************************************************/

void PrintIfNPositive ( void )

/**********************************************************************/
/*
  Purpose:

    PRINTLFNPOSITIVE warns if a discrepancy occurred one or more times.
*/
{
  if ( N > 0 ) 
  {
    printf("Similar discrepancies have occurred %d times.\n", N);
  }
  return;
}
/**********************************************************************/

void TstPtUf ( void )

/**********************************************************************/
/*
  Purpose:

    TstPtUf???
*/
{
  N = 0;
  if ( Z != Zero )
  {
    printf("Since comparison denies Z = 0, evaluating ");
    printf("(Z + Z) / Z should be safe.\n");
    sigsave = sigfpe;
    if (setjmp(ovfl_buf)) goto very_serious;
    Q9 = (Z + Z) / Z;
    printf("What the machine gets for (Z + Z) / Z is  %.17e .\n",
      Q9);
    if (FABS(Q9 - Two) < Radix * U2)
    {
      printf("This is O.K., provided Over/Underflow");
      printf(" has NOT just been signaled.\n");
    }
    else
    {
      if ((Q9 < One) || (Q9 > Two))
      {
very_serious:
        N = 1;
        ErrCnt [Serious] = ErrCnt [Serious] + 1;
        printf("This is a VERY SERIOUS DEFECT!\n");
      }
      else
      {
        N = 1;
        ErrCnt [Defect] = ErrCnt [Defect] + 1;
        printf("This is a DEFECT!\n");
      }
    }
    sigsave = 0;
    V9 = Z * One;
    Random1 = V9;
    V9 = One * Z;
    Random2 = V9;
    V9 = Z / One;
    if ((Z == Random1) && (Z == Random2) && (Z == V9))
    {
      if (N > 0) Pause();
      }
    else {
      N = 1;
      BadCond(Defect, "What prints as Z = ");
      printf("%.17e\n\tcompares different from  ", Z);
      if (Z != Random1) printf("Z * 1 = %.17e ", Random1);

      if (! ( (Z == Random2) || (Random2 == Random1) ) )
      {
        printf("1 * Z == %g\n", Random2);
      }

      if (! (Z == V9)) 
      {
        printf("Z / 1 = %.17e\n", V9);
      }

      if (Random2 != Random1)
      {
        ErrCnt [Defect] = ErrCnt [Defect] + 1;
        BadCond(Defect, "Multiplication does not commute!\n");
        printf("\tComparison alleges that 1 * Z = %.17e\n",
          Random2);
        printf("\tdiffers from Z * 1 = %.17e\n", Random1);
      }
      Pause();
    }
  }
  return;
}
/**********************************************************************/

void notify ( char* s)

/**********************************************************************/
/*
  Purpose:

    NOTIFY warns if a test appears to be inconsistent.
*/
{
  printf("%s test appears to be inconsistent...\n", s);
  printf("   PLEASE NOTIFY KARPINKSI!\n");

  return;
}
/**********************************************************************/

void msglist ( char** s )

/**********************************************************************/
/*
  Purpose:

    MSGLIST prints out a block of test.
*/
{ 
  while ( *s ) 
  {
    printf("%s\n", *s++); 
  }

  return;
}
/**********************************************************************/

void Instructions ( void )

/**********************************************************************/
/*
  Purpose:

    INSTRUCTIONS prints some instructions.
*/
{
  static char *instr[] = {
  "Lest this program stop prematurely, i.e. before displaying\n",
  "    `END OF TEST',\n",
  "try to persuade the computer NOT to terminate execution when an",
  "error like Over/Underflow or Division by Zero occurs, but rather",
  "to persevere with a surrogate value after, perhaps, displaying some",
  "warning.  If persuasion avails naught, don't despair but run this",
  "program anyway to see how many milestones it passes, and then",
  "amend it to make further progress.\n",
  "Answer questions with Y, y, N or n (unless otherwise indicated).\n",
  0};

  msglist(instr);

  return;
}
/**********************************************************************/

void Heading ( void )

/**********************************************************************/
/*
  Purpose:

    HEADING prints a heading.
*/
{
  static char *head[] = 
  {
    "Users are invited to help debug and augment this program so it will",
    "cope with unanticipated and newly uncovered arithmetic pathologies.\n",
    "Please send suggestions and interesting results to",
    "\tRichard Karpinski",
    "\tComputer Center U-76",
    "\tUniversity of California",
    "\tSan Francisco, CA 94143-0704, USA\n",
    "In doing so, please include the following information:",
#ifdef Single
    "\tPrecision:\tsingle;",
#else
    "\tPrecision:\tdouble;",
#endif
    "\tVersion:\t10 February 1989;",
    "\tComputer:\n",
    "\tCompiler:\n",
    "\tOptimization level:\n",
    "\tOther relevant compiler options:",
    0
  };

  msglist ( head );
  return;
}
/**********************************************************************/

void Characteristics ( void )

/**********************************************************************/
/*
  Purpose:

    CHARACTERISTICS explains the arithmetic properties to be checked.
*/
{
  static char *chars[] = 
  {
    "Running this program should reveal these characteristics:",
    "     Radix = 1, 2, 4, 8, 10, 16, 100, 256 ...",
    "     Precision = number of significant digits carried.",
    "     U2 = Radix/Radix^Precision = One Ulp",
    "\t(OneUlpnit in the Last Place) of 1.000xxx .",
    "     U1 = 1/Radix^Precision = One Ulp of numbers a little less than 1.0 .",
    "     Adequacy of guard digits for Mult., Div. and Subt.",
    "     Whether arithmetic is chopped, correctly rounded, or something else",
    "\tfor Mult., Div., Add/Subt. and Sqrt.",
    "     Whether a Sticky Bit used correctly for rounding.",
    "     UnderflowThreshold = an underflow threshold.",
    "     E0 and PseudoZero tell whether underflow is abrupt, gradual, or fuzzy.",
    "     V = an overflow threshold, roughly.",
    "     V0  tells, roughly, whether  Infinity  is represented.",
    "     Comparisions are checked for consistency with subtraction",
    "\tand for contamination with pseudo-zeros.",
    "     Sqrt is tested.  Y^X is not tested.",
    "     Extra-precise subexpressions are revealed but NOT YET tested.",
    "     Decimal-Binary conversion is NOT YET tested for accuracy.",
    0
  };

  msglist ( chars );

  return;
}
/**********************************************************************/

void History ( void )

/**********************************************************************/
{ 
/*
  Purpose:

    HISTORY prints out an explanation of the program, and history.

  Discussion:

    Converted from Brian Wichmann's Pascal version to C by Thos Sumner
    with further massaging by David M. Gay
*/
  static char *hist[] = 
  {
    "The program attempts to discriminate among",
    "   FLAWs, like lack of a sticky bit,",
    "   Serious DEFECTs, like lack of a guard digit, and",
    "   FAILUREs, like 2+2 == 5 .",
    "Failures may confound subsequent diagnoses.\n",
    "The diagnostic capabilities of this program go beyond an earlier",
    "program called `MACHAR', which can be found at the end of the",
    "book  `Software Manual for the Elementary Functions' (1980) by",
    "W. J. Cody and W. Waite. Although both programs try to discover",
    "the Radix, Precision and range (over/underflow thresholds)",
    "of the arithmetic, this program tries to cope with a wider variety",
    "of pathologies, and to say how well the arithmetic is implemented.",
    "\nThe program is based upon a conventional radix representation for",
    "floating-point numbers, but also allows logarithmic encoding",
    "as used by certain early WANG machines.\n",
    "BASIC version of this program (C) 1983 by Prof. W. M. Kahan;",
    "see source comments for more history.",
    0
  };

  msglist ( hist );

  return;
}
